Journal of Articial Intelligence Research Submitted published TotalOrder and PartialOrder Planning A Comparative Analysis Steven Minton mintonptolemyarcnasagov John Bresina bresinaptolemyarcna sagov Mark Drummond medptolemyarcnasagov Recom Technologies NASA Ames Research Center Mail Stop Moett Field CA USA Abstract For many years the intuitions underlying partialorder planning were largely taken for granted Only in the past few years has there b een renewed interest in the fundamental principles underlying this paradigm In this pap er we present a rigorous comparative analysis of partialorder and totalorder planning by fo cusing on two sp ecic planners that can b e directly compared We show that there are some subtle assumptions that underly the widespread intuitions regarding the supp osed eciency of partialorder planning For instance the sup eriority of partialorder planning can dep end critically up on the search strategy and the structure of the search space Understanding the underlying assumptions is crucial for constructing ecient planners Intro duction For many years the sup eriority of partialorder planners over totalorder planners has b een tacitly assumed by the planning community Originally partialorder planning was intro duced by Sacerdoti as a way to improve planning eciency by avoiding premature commitments to a particular order for achieving subgoals The utility of partialorder planning was demonstrated anecdotally by showing how such a planner could eciently solve blo cksworld examples such as the wellknown Sussman anomaly Since partialorder planning intuitively seems like a go o d idea little attention has b een devoted to analyzing its utility at least until recently Minton Bresina Drummond a Barrett Weld Kambhampati c However if one lo oks closely at the issues involved a numb er of questions arise For example do the advantages of partial order planning hold regardless of the search strategy used Do the advantages hold when the planning language is so expressive that reasoning ab out partially ordered plans is intractable eg if the language allows conditional eects Our work Minton et al a has shown that the situation is much more inter esting than might b e exp ected We have found that there are some unstated assumptions underlying the supp osed eciency of partialorder planning For instance the sup eriority of partialorder planning can dep end critically up on the search strategy and search heuristics employed This pap er summarizes our observations regarding partialorder and totalorder plan ning We b egin by considering a simple totalorder planner and a closely related partial order planner and establishing a mapping b etween their search spaces We then examine c AI Access Foundation and Morgan Kaufmann Publishers All rights reserved Minton Bresina Drummond the relative sizes of their search spaces demonstrating that the partialorder planner has a fundamental advantage b ecause the size of its search space is always less than or equal to that of the totalorder planner However this advantage do es not necessarily translate into an eciency gain this dep ends on the typ e of search strategy used For example we describ e a domain where our partial order planner is more ecient than our total order planner when depthrst search is used but the eciency gain is lost when an iterative sampling strategy is used We also show that partialorder planners can have a second indep endent advantage when certain typ es of op erator ordering heuristics are employed This heuristic advantage underlies Sacerdotis anecdotal examples explaining why leastcommitment works However in our blo cksworld exp eriments this second advantage is relatively unimp ortant compared to the advantage derived from the reduction in search space size Finally we lo ok at how our results extend to partialorder planners in general We describ e how the advantages of partialorder planning can b e preserved even if highly ex pressive languages are used We also show that the advantages do not necessarily hold for all partialorder planners but dep end critically on the construction of the planning space Background Planning can b e characterized as search through a space of p ossible plans A totalorder planner searches through a space of totally ordered plans a partialorder planner is dened analogously We use these terms rather than the terms linear and nonlinear b ecause the latter are overloaded For example some authors have used the term nonlinear when fo cusing on the issue of goal ordering That is some linear planners when solving a conjunctive goal require that all subgoals of one conjunct b e achieved b efore subgoals of the others hence planners that can arbitrarily interleave subgoals are often called nonlinear This version of the linearnonlinear distinction is dierent than the partialordertotal order distinction investigated here The former distinction impacts planner completeness whereas the totalorderpartialorder distinction is orthogonal to this issue Drummond Currie Minton et al a The totalorderpartialorder distinction should also b e kept separate from the distinc tion b etween worldbased planners and planbased planners The distinction is one of mo deling in a worldbased planner each search state corresp onds to a state of the world and in a planbased planner each search state corresp onds to a plan While total order planners are commonly asso ciated with worldbased planners such as Strips several wellknown totalorder planners have b een planbased such as Waldingers regression plan ner Waldinger Interplan Tate and Warplan Warren Similarly partialorder planners are commonly planbased but it is p ossible to have a worldbased partialorder planner Go defroid Kabanza In this pap er we fo cus solely on the totalorderpartialorder distinction in order to avoid complicating the analysis We claim that the only signicant dierence b etween partialorder and totalorder plan ners is planning eciency It might b e argued that partialorder planning is preferable b ecause a partially ordered plan can b e more exibly executed However execution exibil ity can also b e achieved with a totalorder planner and a p ostpro cessing step that removes unnecessary orderings from the totally ordered solution plan to yield a partial order Back TotalOrder and PartialOrder Planning strom Veloso Perez Carb onell Regnier Fade The p olynomial time complexity of this p ostpro cessing is negligible compared to the search time for plan generation Hence we b elieve that execution exibility is at b est a weak justication for the supp osed sup eriority of partialorder planning In the following sections we analyze the relative eciency of partialorder and total order planning by considering a totalorder planner and a partialorder planner that can b e directly compared Elucidating the key dierences b etween these planning algorithms reveals some imp ortant principles that are of general relevance Terminology A plan consists of an ordered set of steps where each step is a unique op erator instance Plans can b e total ly ordered in which case every step is ordered with resp ect to every other step or partial ly ordered in which case steps can b e unordered with resp ect to each other We assume that a library of op erators is available where each op erator has preconditions deleted conditions and added conditions All of these conditions must b e nonnegated prop o sitions and we adopt the common convention that each deleted condition is a precondition Later in this pap er we show how our results can b e extended to more expressive languages but this simple language is sucient to establish the essence of our argument A linearization of a partially ordered plan is a total order over the plans steps that is consistent with the existing partial order In a totally ordered plan a precondition of a plan step is true if it is added by an earlier step and not deleted by an intervening step In a partially ordered plan a steps precondition is possibly true if there exists a linearization in which it is true and a steps precondition is necessarily true if it is true in al l linearizations A steps precondition is necessarily false if it is not p ossibly true A state consists of a set of prop ositions A planning problem is dened by an initial state and a set of goals where each goal is a prop osition For convenience we represent a problem as a twostep initial plan where the prop ositions that are true in the initial state are added by the rst step and the goal prop ositions are the preconditions of the nal step The planning pro cess starts with this initial plan and searches through a space of p ossible plans A successful search terminates with a solution plan ie a plan in which all steps preconditions are necessarily true The search space can b e characterized as a tree where each no de corresp onds to a plan and each arc corresp onds to a plan transformation Each transformation incrementally extends ie renes a plan by adding additional steps or orderings Thus each leaf in the search tree corresp onds either to a solution plan or a deadend and each intermediate no de corresp onds to an unnished plan which can b e further extended Backstrom formalizes the problem of removing unnecessary orderings in order to pro duce a least constrained plan He shows that the problem is p olynomial if one denes a leastconstrained plan as a plan in which no orderings can b e removed without impacting the correctness of the plan Backstrom also shows that the problem of nding a plan with